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 Jul23 comment What are some conceptualizations that work in mathematics but are not strictly true? @Semiclassical The notion of $\theta^2$ gives me a bit of a headache. Apr21 awarded Commentator Apr21 comment Why is P or not P is unsatisifiable by construction? As for "by construction": the phrase has two related meanings. A "proof by construction" is a proof that asserts an object exists by constructing it. The sentence "By construction, X has property Y" means "the way in which X was described (i.e., its construction) makes it easy to see that it has property Y". Apr21 comment Why is P or not P is unsatisifiable by construction? You're reading an 'or' in the formula where you should be reading an 'and'. In order for a set such as $\Delta$ to be satisfied by a model, every formula in the set has to be satisfied simultaneously by that model. For any model $G$, if $G$ satisfies $\phi[c/u][c'/v]$, then there is some $n$ such that $G$ does not satisfy $\neg \phi_n$. Therefore, it cannot satisfy $\Delta$, since $\neg \phi_n \in \Delta$. Apr20 answered Why is P or not P is unsatisifiable by construction? Apr3 comment Con(PA) implies consistency of $\mathsf{PA}$ + ¬Con($\mathsf{PA}$) The one thing I'm a little unclear is what you mean by 'completeness'; if PA is consistent, doesn't that necessarily mean that it's incomplete? Apr3 accepted Con(PA) implies consistency of $\mathsf{PA}$ + ¬Con($\mathsf{PA}$) Apr2 asked Con(PA) implies consistency of $\mathsf{PA}$ + ¬Con($\mathsf{PA}$) Aug21 awarded Good Question Feb16 awarded Notable Question Apr30 accepted Existence of a particular well-ordering of [0,1] Apr30 comment Existence of a particular well-ordering of [0,1] Ah, ok. Thanks! Apr30 comment Existence of a particular well-ordering of [0,1] So I can see how you need CH to show that $\omega_1$ can be bijected with $\mathbb{R}$ (otherwise you just know that $\omega_1$ is uncountable), but I'm not sure how AC comes into play. Apr30 comment Existence of a particular well-ordering of [0,1] I'm familiar with the 'basic' infinite ordinals like $\omega$, less so with $\omega_1$. Apr30 comment Existence of a particular well-ordering of [0,1] Could you explain further? Apr30 asked Existence of a particular well-ordering of [0,1] Apr26 awarded Critic Apr26 comment Eccentricity of an ellipse What is C? What is A? Apr26 awarded Supporter Apr25 awarded Yearling